Deriving the Quadratic Formula by Completing the Square

Oct 13, 2025 by windailys.com 10 views

Introduction

The quadratic formula is a fundamental concept in algebra, used to solve quadratic equations of the form $ax^2 + bx + c = 0$ . While the quadratic formula can be derived using various methods, one of the most popular and intuitive approaches is completing the square. In this article, we will explore the process of deriving the quadratic formula by completing the square and identify the expression that can be added to both sides of the equation to create a perfect square trinomial.

Completing the Square: A Brief Overview

Completing the square is a technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This involves adding and subtracting a constant term to create a perfect square trinomial, which can then be factored into the square of a binomial. The general form of a quadratic expression is $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Deriving the Quadratic Formula by Completing the Square

To derive the quadratic formula by completing the square, we start with the general form of a quadratic equation: $ax^2 + bx + c = 0$ . Our goal is to rewrite this equation in the form of a perfect square trinomial, which can then be factored into the square of a binomial.

Step 1: Divide Both Sides by $a$

The first step in completing the square is to divide both sides of the equation by $a$ , the coefficient of the quadratic term. This gives us:

x^2 + \frac{b}{a} x + \frac{c}{a} = 0

Step 2: Add and Subtract $\left(\frac{b}{2a}\right)^2$

To create a perfect square trinomial, we need to add and subtract a constant term that is equal to the square of half the coefficient of the linear term. In this case, the coefficient of the linear term is $\frac{b}{a}$ , so we add and subtract $\left(\frac{b}{2a}\right)^2$ :

x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} = 0

Step 3: Factor the Perfect Square Trinomial

The next step is to factor the perfect square trinomial on the left-hand side of the equation:

\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} = 0

Step 4: Simplify the Equation

We can simplify the equation by combining like terms:

\left(x + \frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2 - \frac{c}{a}

Step 5: Take the Square Root of Both Sides

To solve for $x$ , we take the square root of both sides of the equation:

x + \frac{b}{2a} = \pm \sqrt{\left(\frac{b}{2a}\right)^2 - \frac{c}{a}}

Step 6: Solve for $x$

Finally, we can solve for $x$ by subtracting $\frac{b}{2a}$ from both sides of the equation:

x = -\frac{b}{2a} \pm \sqrt{\left(\frac{b}{2a}\right)^2 - \frac{c}{a}}

The Quadratic Formula

The final expression for $x$ is the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Conclusion

In this article, we have derived the quadratic formula by completing the square. We started with the general form of a quadratic equation and used the technique of completing the square to rewrite it in the form of a perfect square trinomial. By factoring the perfect square trinomial and simplifying the equation, we arrived at the quadratic formula. The expression that can be added to both sides of the equation to create a perfect square trinomial is $\left(\frac{b}{2a}\right)^2$ .

Answer

$The correct answer is **B. $\frac{c}{2a}$**.<br/> **Quadratic Formula Q&A: Frequently Asked Questions**$

Introduction

The quadratic formula is a powerful tool used to solve quadratic equations of the form $ax^2 + bx + c = 0$ . In our previous article, we derived the quadratic formula by completing the square. In this article, we will answer some of the most frequently asked questions about the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$ . It is given by:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. Then, simplify the expression and solve for $x$ .

Q: What is the significance of the quadratic formula?

A: The quadratic formula is a fundamental concept in algebra and is used to solve quadratic equations. It is a powerful tool that can be used to solve a wide range of problems in mathematics, science, and engineering.

Q: Can I use the quadratic formula to solve quadratic equations with complex roots?

A: Yes, the quadratic formula can be used to solve quadratic equations with complex roots. If the discriminant ( $b^2 - 4ac$ ) is negative, then the quadratic equation has complex roots.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you need to look at the discriminant ( $b^2 - 4ac$ ). If the discriminant is:

Positive, then the quadratic equation has two distinct real solutions.
Zero, then the quadratic equation has one real solution.
Negative, then the quadratic equation has two complex solutions.

Q: Can I use the quadratic formula to solve quadratic equations with rational coefficients?

A: Yes, the quadratic formula can be used to solve quadratic equations with rational coefficients. However, you need to be careful when simplifying the expression, as you may end up with complex roots.

Q: How do I simplify the quadratic formula?

A: To simplify the quadratic formula, you need to follow the order of operations (PEMDAS):

Evaluate the expression inside the square root.
Simplify the expression under the square root.
Simplify the expression on top of the square root.
Simplify the final expression.

Q: Can I use the quadratic formula to solve quadratic equations with non-integer coefficients?

A: Yes, the quadratic formula can be used to solve quadratic equations with non-integer coefficients. However, you need to be careful when simplifying the expression, as you may end up with complex roots.

Conclusion

In this article, we have answered some of the most frequently asked questions about the quadratic formula. We hope that this article has been helpful in clarifying any doubts you may have had about the quadratic formula.

Frequently Asked Questions

What is the quadratic formula?
How do I use the quadratic formula?
What is the significance of the quadratic formula?
Can I use the quadratic formula to solve quadratic equations with complex roots?
How do I determine the number of solutions to a quadratic equation?
Can I use the quadratic formula to solve quadratic equations with rational coefficients?
How do I simplify the quadratic formula?
Can I use the quadratic formula to solve quadratic equations with non-integer coefficients?

Answers

The quadratic formula is a mathematical formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$ .
To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ into the formula and simplify the expression.
The quadratic formula is a fundamental concept in algebra and is used to solve quadratic equations.
Yes, the quadratic formula can be used to solve quadratic equations with complex roots.
To determine the number of solutions to a quadratic equation, you need to look at the discriminant ( $b^2 - 4ac$ ).
Yes, the quadratic formula can be used to solve quadratic equations with rational coefficients.
To simplify the quadratic formula, you need to follow the order of operations (PEMDAS).
Yes, the quadratic formula can be used to solve quadratic equations with non-integer coefficients.